An electromagnetic wave of v=3 MHz passes from vacuum into dielectric medium with` epsi= 4 epsi_0`.
then

Options

A. Wavelength is doubled and frequency becomes half .
B. Wavelength is doubled and frequency is same .
C. Wavelength and frequency both remain unchanged .
D. Wavelength is halved but frequency remains same .

Explanation

To solve the problem regarding an electromagnetic wave passing from vacuum into a dielectric medium, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Frequency of the electromagnetic wave, \( f = 3 \, \text{MHz} \).
- Permittivity of the dielectric medium, \( \epsilon = 4 \epsilon_0 \).

2. **Understand the Properties of Electromagnetic Waves**:
- When an electromagnetic wave travels from one medium to another, the frequency remains unchanged. Thus, the frequency in the dielectric medium is still \( f = 3 \, \text{MHz} \).

3. **Determine the Speed of the Wave in Different Media**:
- The speed of light in vacuum is given by \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \).
- The speed of the wave in the dielectric medium can be calculated using the formula:
\[ v = \frac{c}{\sqrt{\epsilon_r}} \]
- Here, \( \epsilon_r \) (relative permittivity) is given by \( \epsilon_r = \frac{\epsilon}{\epsilon_0} = 4 \).

4. **Calculate the Speed in the Dielectric Medium**:
- Substituting \( \epsilon_r \) into the speed formula:
\[ v = \frac{c}{\sqrt{4}} = \frac{c}{2} \]
- Therefore, the speed of the wave in the dielectric medium is \( \frac{c}{2} \).

5. **Relate Wavelength, Frequency, and Speed**:
- The relationship between speed, frequency, and wavelength is given by:
\[ v = f \lambda \]
- In vacuum, the wavelength \( \lambda_1 \) can be expressed as:
\[ \lambda_1 = \frac{c}{f} \]
- In the dielectric medium, the wavelength \( \lambda_2 \) can be expressed as:
\[ \lambda_2 = \frac{v}{f} = \frac{\frac{c}{2}}{f} = \frac{c}{2f} \]

6. **Calculate the Relationship Between Wavelengths**:
- Now, we can find the ratio of the wavelengths:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\frac{c}{f}}{\frac{c}{2f}} = \frac{c}{f} \times \frac{2f}{c} = 2 \]
- This indicates that \( \lambda_1 = 2 \lambda_2 \), meaning the wavelength in the dielectric medium is half that in vacuum.

7. **Conclusion**:
- The frequency remains unchanged at \( 3 \, \text{MHz} \).
- The wavelength in the dielectric medium is half that in vacuum.

### Final Answer:
- The wavelength is halved and the frequency remains the same.